derivative of (sqrt(x)/(1+x^{3/2)})

Lösung

\frac{d}{d x} (\frac{x}{1 + x ^{\frac{3}{2}}})

\frac{1 + x ^{\frac{3}{2}} - 3 x x}{2 x ^{\frac{1}{2}} ( 1 + x ^{\frac{3}{2}} ) ^{2}}

Schritte zur Lösung

\frac{d}{d x} (\frac{x}{1 + x ^{\frac{3}{2}}})

Wende die Quotientenregel an:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( x ) ( 1 + x ^{\frac{3}{2}} ) - \frac{d}{d x} ( 1 + x ^{\frac{3}{2}} ) x}{( 1 + x ^{\frac{3}{2}} ) ^{2}}

\frac{d}{d x} (x) = \frac{1}{2 x ^{\frac{1}{2}}}

\frac{d}{d x} (1 + x^{\frac{3}{2}}) = \frac{3 x ^{\frac{1}{2}}}{2}

= \frac{\frac{1}{2 x ^{\frac{1}{2}}} ( 1 + x ^{\frac{3}{2}} ) - \frac{3 x ^{\frac{1}{2}}}{2} x}{( 1 + x ^{\frac{3}{2}} ) ^{2}}

Vereinfache

\frac{\frac{1}{2 x ^{\frac{1}{2}}} ( 1 + x ^{\frac{3}{2}} ) - \frac{3 x ^{\frac{1}{2}}}{2} x}{( 1 + x ^{\frac{3}{2}} ) ^{2}} : \frac{1 + x ^{\frac{3}{2}} - 3 x x}{2 x ^{\frac{1}{2}} ( 1 + x ^{\frac{3}{2}} ) ^{2}}

= \frac{1 + x ^{\frac{3}{2}} - 3 x x}{2 x ^{\frac{1}{2}} ( 1 + x ^{\frac{3}{2}} ) ^{2}}

\frac{\partial}{\partial x} (\frac{y}{x})

\int_{0}^{x} t^{2} + t^{4} d t

\frac{d y}{d x} = y^{2} - 4

\frac{\partial}{\partial x} (sin (5 π x + 2 y))

\frac{d}{d x} (\frac{x y ( x ^{2} - y ^{2} )}{x ^{2} + y ^{2}})